Multiple bifurcations of a time-delayed coupled FitzHugh–Rinzel neuron system with chemical and electrical couplings

In this paper, a time-delayed coupled FitzHugh–Rinzel neuron system (two neurons) with electrical and chemical couplings is discussed. First, when choosing different strengths of electrical (or chemical) coupling, the number and position of equilibria of the neuron system without time delays affected by the variation of chemical (or electrical) coupling are investigated in detail. Next, based on the discussion about the equilibria of the coupled neuron system without time delays, the codimension one bifurcations of equilibria or limit cycles including static bifurcation, Hopf bifurcation and fold bifurcation of limit cycles are deduced when choosing the strength of electrical coupling or chemical coupling as the bifurcation parameter, respectively. Furthermore, the codimension two bifurcations of equilibria are discussed which exhibit more fascinating and complicated dynamical behaviors. Synchronization problems influenced by the strength of electrical coupling are also considered which contain the complete synchronous and asynchronous state. After the discussion of the neuron system without time delays, the impact of time delays on the stability of symmetric equilibria of coupled FitzHugh–Rinzel neuron system is explored. The phenomenon of stability switching by a Hopf bifurcation is well detected. Delay-dependent Hopf curves are found and the dynamical behaviors near the intersection point of Hopf bifurcation (Hopf–Hopf bifurcation point) in the parameter plane of two time delays are investigated where there exists the coexistence of two limit cycles with different frequencies. The numerical simulations are exhibited after the discussion of each part.